Deterministic computations whose history is independent of the order of asynchronous updating

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state ξ(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is η(x , t) then let us define the sequence ζ(x , 0),ζ(x , 1), . . . by taking the sequence η(x , 0),η(x , 1), . . . , and deleting each repetition, i.e. each element equal to the preceding one. The function f is said to have invariant histories if the sequence ζ(x , i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x , y of neighbors, if the updating would change both ξ(x) and ξ(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order. This fact is related to the confluence property of termrewriting systems and well as sandpile theory in statistical physics. We will also show a simple simulation of an arbitrary synchronous computation by a commutative asynchronous one. ∗Partially supported by NSF grant CCR-920484